This sounds like the possible beginning of encopresis. They don't technically diagnosis it until after age four though. That is WAY too long to hold poop. I am sure that she is bound to have it backed way up. She may very well likely have stretched her colon and is losing the natural urge to go.
We have had pooping problems with our 3yo since shortly after he turned 2 (but the most he ever held it back was probably 5-6 days.) I was really scared it would turn into encopresis but luckily he is finally healing! His problem was fructose malabsorption. The FM gave him acidic poop that burned so he held it back. Once we fixed the diet issues he started to heal physically and emotionally. And oh boy did that take dedication, perseverance and a complete change in our diets - we journaled every morsel he ate for months. We're now wheat/gluten free (wheat has fructans and we just haven't tried putting gluten back yet), low on lactose (kids with FM are often also lactose intolerant) and completely fruit (and tomato) free. He eats no honey, molasses, brown sugar, etc. He can handle veggies with good glucose/fructose ratios in moderation, sucrose and evaporated cane juice in moderation. Our diet is pretty limited but we're slowly expanding. We have minor setbacks at times when we get a little too much fructose but it isn't enough to affect his healing progress.
When this first started with him - we thought his episodes of "holding" back may be behavior-oriented and that holding the poop back somehow caused the sore bottom when it finally came out. (Totally wrong!)
A ped suggested Miralax but we didn't understand how that would work given he wasn't constipated and then when I researched it, I definitely ruled it out (find the Yahoo group about Miralax side effects - scary!) For a while, I gave him prune juice, pears and flax oil to keep his bowels moving (Totally wrong!)
which was ultimately only making the problem worse.
We have a very long and detailed story that I can't even begin to fully express here. I do know, however, that I have read many stories in my research about how these issues can ultimately be diet related.
If I were you, I'd start food journaling and consider elimination diets. Unfortunately, you are likely in for a very long and arduous journey regardless of the cause. These issues take a long time to heal. It took about 3 months for our son's problem to escalate (as he started eating more fructose), about 3 months more until we were doing elimination diets and about 4 more months for us to adjust his diet and FINALLY figure out it was fructose. It was baby steps all the way with healing. While he was still eating a ton of fructans and fructose, he would poop in his diaper after days of holding it back but only if we picked him up and hugged him. When we cut wheat and some other sources of fructose (without knowing it was the fructose causing the issue) he started to poop in the potty but it took a long time - we had to sit with him for long periods of time and play with toys. As I started to figure out what foods and quantities he tolerated (still didn't realize it was fructose but I knew he couldn't eat too much of this or that) he started pooping in the potty more quickly. And when I finally put all the pieces of the puzzle together and made the fructose connection, he finally went to the potty on his own. You have no idea how insanely ecstatic I was the first time he went completely on his own. I felt like I had finally won a long uphill battle.
Overall though, our stories may be completely different but I did want to put the diet suggestion out there. I joined lists about encopresis and miralax and frequented the allergy forums here. I found lots of tips and ideas but ultimately, I found not one other person who had identified the problem we had. I would seek out these groups though - if her problem is functional constipation (which is the cause of many cases of encopresis), there were many families who have an arsenal of natural products to help (fruit-eze, oils, magnesium, prune juice, pears, etc.) However, many have also found that the constipation went away when they removed something from the diet (like gluten). There are also people who've had luck with behavior therapies. You may have some detective work ahead of you and the problem may persist to some degree despite your best effort. I wish the best of luck; I know how incredibly difficult this is to deal with.
ETA: I wanted to add though, if she hasn't gone in three weeks she is going to need some form of clean-out therapy if you can't get her to go on her own. The runny poo coming out is likely liquid stool coming out around the hard stool built up inside. I would be pretty worried at this point.
I hope she goes soon.
